Lexicographic and colexicographic order

Lex order is that of a dictionary.
CoLex order is obtained by reflecting all tuples, applying Lex order, and reflecting the tuples again.

Lex order is more intuitive for most people.
CoLex order is more practical when the finite sets of tuples to be ordered shall be generalized to infinite sets of sequences.

Both orderings can be reversed, so there are actually four different orderings.
They can also be reflected (see here), but that's not a different ordering of the set of tuples, but just a certain ordering written in a different way.
When the set of tuples contains all reflections, each reflected order is equal to some of the four others (see below).

Contents

The sequence of the (nk){\displaystyle \scriptstyle {\binom {n}{k}}}k-subsets of {1,...,n}{\displaystyle \scriptstyle \{1,...,n\}} in CoLex order
is the beginning of the infinite sequence of k-subsets of {1,2,3,...}{\displaystyle \scriptstyle \{1,2,3,...\}} in CoLex order.
This corresponds to the increasing sequence A014311 = 7,11,13,14,19,21...

In the file on the left it can be seen that the blue patterns are horizontally reflected.
However, this is not the case in the file on the right, which shows only some of the subsets.

The 10 3-subsets of {1,...,6}{\displaystyle \scriptstyle \{1,...,6\}} with an even sum of elements

The (53)=10{\displaystyle \scriptstyle {\binom {5}{3}}=10} 3-subsets of {1,...,5}{\displaystyle \scriptstyle \{1,...,5\}}(binary vectors RevLex, combinations Lex)
This sequence of combinations is not the beginning of the sequence of combinations in Lex order in the file on the left.

The triangle on the right shows 2-subsets of the integers.
This corresponds to the sequence A018900as a square array.
The sequence of numbers (3,5,6,9,10,12...) corresponds to the CoLex ordering of 2-subsets:

Permutations are often shown in Lex order (see here),
but RevCoLex is the order that works for an infinite number of permutations
and corresponds to the CoLex order of the permutations' reflected factorial numbers (compare Inversion).

The 24 permutations of {1,...,5}{\displaystyle \scriptstyle \{1,...,5\}}
that have a complete 5-cycle

The 12 even permutations of {1,...,4}{\displaystyle \scriptstyle \{1,...,4\}}(the ones that are not green or orange in the table on the right)

Subsets ordered with the same pattern, but reflected
(This order does not seem to have a name.)

The following Python code shows the 16 subsets of the set {a,b,c,d} in lexicographic order.
The second part shows that Lex order is the same as binary CoLex order, when the subsets themselves are reversed.